1. Evaluate nth Roots and Use Rational Exponents
Chapter 6.1

2. Nth Roots
In an earlier chapter, we defined the square root of a number as follows If 𝑥 2 =𝑎, then 𝑥=± 𝑎
When solving for a square, we can ask, “What number multiplies itself to produce a?”
For example, if 𝑥 2 =4, then x is the number that multiplies itself to produce 4
It turns out that there are actually two such numbers: 2 and −2, because 2 2 =4 and −2 2 =−2⋅−2=4

3. Nth Roots
We can extend this idea further by asking of some number a, “What multiplies itself three times to produce a?”
We can express this as 𝑥 3 =𝑎
For example, what number multiplies itself three times to produce 8?
This is the solution to the equation 𝑥 3 =8
Since 2⋅2⋅2= 2 3 =8, then 𝑥=2; note that there is only one solution
We can define the cube root of a number as If 𝑥 3 =𝑎, then 𝑥= 3 𝑎

4. Nth Roots
We can define the cube root of a number as If 𝑥 3 =𝑎, then 𝑥= 3 𝑎
Notice that, unlike a square root, a cube root has only one answer
Also unlike a square root, the cube root of any real number is a real number
Recall that if 𝑥 2 =−1, then x is not a real number; generally, if 𝑥 2 equals any negative number, the result is imaginary
But if 𝑥 3 =−8, then 𝑥= 3 −8 =−2 because −2⋅−2⋅−2=−8

5. Nth Roots Can we extend this further to 4th roots?
If 𝑥 4 =𝑎, then x is the number that multiplies itself four times to produce a
For example, if 𝑥 4 =81, then 𝑥=3 because 3⋅3⋅3⋅3= 3⋅3 ⋅ 3⋅3 =9⋅9=81
But x can also be −3 because −3⋅−3 ⋅ −3⋅−3 =9⋅9=81
So we define a fourth root as
If 𝑥 4 =𝑎, then 𝑥=± 4 𝑎

6. Nth Roots So we define a fourth root as If 𝑥 4 =𝑎, then 𝑥=± 4 𝑎
As will square roots, the fourth root of a negative number is imaginary
For example, 𝑥 4 =−16 has no real solution because there is no real number that multiplies itself 4 times to produce −16
We will only be concerned with real numbers in this chapter, so we will say that this has no real solution

7. Nth Roots For the notation 𝑛 𝑎 we would say “the nth root of a”
The whole number n is called the index of the radical and a is again called the radicand
Only two roots have special names: square root and cube root
Also, the index for a square root is not written; that is 2 𝑎 = 𝑎
In defining nth roots, we must differentiate between even and odd roots

8. n is an even whole number
Nth Roots
Suppose that n is a whole number greater than 1, and that a is a real number such that 𝑥 𝑛 =𝑎
n is an even whole number
n is an odd whole number
If 𝑎<0, there are no real nth roots
If 𝑎<0, then 𝑥= 𝑛 𝑎 (1 solution, negative)
If 𝑎=0, then 𝑥= 𝑛 0 =0 (1 solution)
If 𝑎>0, then 𝑥=± 𝑛 𝑎 (2 solutions)
If 𝑎>0, then 𝑥= 𝑛 𝑎 (1 solution, positive)

9. Guided Practice
Use the definition of nth roots and your calculator to find x.
𝑥 5 =1024
𝑥 4 =16
𝑥 3 =−125
𝑥 6 =−1
𝑥 5 =2

10. Guided Practice
Use the definition of nth roots and your calculator to find x.
𝑥 5 =1024; 𝑥= =4 because 4⋅4⋅4⋅4⋅4=1024
𝑥 4 =16; 𝑥= 4 16 =2 because 2⋅2⋅2⋅2=16
𝑥 3 =−125; 𝑥= 3 −125 =−5 because −5⋅−5⋅−5=−125
𝑥 6 =−1; 𝑥= 6 −1 , but since the radicand is negative, there is no real solution
𝑥 5 =2; 𝑥= 5 2 ≈ because (1.1487)(1.1487)(1.1487)(1.1487)(1.1487)≈2

11. Rational Exponents Up to now we have only used integer exponents
Recall that 𝑎 𝑛 =𝑎⋅𝑎⋅…⋅𝑎, where there are n factors of a
Also, 𝑎 −𝑛 = 1 𝑎 𝑛 = 1 𝑎 ⋅ 1 𝑎 ⋅…⋅ 1 𝑎 , where there are n factors of 1 𝑎
We can extend the meaning of an exponent to rational numbers (that is, to fractions)
To do this we must recall one of the properties of exponents: The power property says that 𝑎 𝑛 𝑚 = 𝑎 𝑛𝑚 (multiply the exponents)
Also recall that 𝑛⋅ 1 𝑛 =1 (inverse property of multiplication)

12. Rational Exponents How can we interpret the equation 𝑥= 25 1 2 ?
Suppose that we raise both sides to the power 2: 𝑥 2 =
By the power property of exponents we get 𝑥 2 = ⋅2
By the inverse property of multiplication, 𝑥 2 = 25 1 =25
But by the definition of square root, 𝑥= 25 =5 (we don’t need both positive and negative here)
Now, since 𝑥= and 𝑥= 25 , then 25 = =5

13. Rational Exponents
More generally, if 𝑥= 𝑎 1 𝑛 , then 𝑥 𝑛 = 𝑎 1 𝑛 𝑛 = 𝑎 1 𝑛 ⋅𝑛 = 𝑎 1 =𝑎
Since 𝑥 𝑛 =𝑎, then by the definition of nth roots 𝑥= 𝑛 𝑎
Now, 𝑥= 𝑎 1 𝑛 and 𝑥= 𝑛 𝑎 , so we conclude that 𝑎 1 𝑛 = 𝑛 𝑎
Rational exponents of the form 1 𝑛 are the same thing as nth roots

14. Rational Exponents
This means that we can rewrite nth roots of a number as the number raised to the 1 𝑛 power, and vice verse
For example, 4 7 = , 3 9 = , and 2 =
Remember that we don’t write the index for the square root
Also, = 6 , = 5 3 , and = 4 10

15. Guided Practice
Rewrite the expression using rational exponent notation.
3 5 =
15 =
8 6 =
𝑛 𝑝 =

16. Guided Practice
Rewrite the expression using rational exponent notation.
3 5 =
15 =
8 6 =
𝑛 𝑝 = 𝑝 1 𝑛

17. Guided Practice Rewrite the expression using radical notation. 4 1 3 == = =
𝑥 1 𝑛 =

18. Guided Practice Rewrite the expression using radical notation.
= 3 4
= 5 11
= 3
𝑥 1 𝑛 = 𝑛 𝑥

19. Rational Exponents With Denominator Other Than 1
Next we must interpret the meaning of 𝑚 𝑛 as an exponent, where m is not 1
We can rewrite 𝑚 𝑛 as 𝑚⋅ 1 𝑛 or as 1 𝑛 ⋅𝑚 (commutative property)
Now, by the power property of exponents 𝑎 1 𝑛 𝑚 = 𝑎 1 𝑛 ⋅𝑚 = 𝑎 𝑚 𝑛
But we also have that 𝑎 1 𝑛 = 𝑛 𝑎 , so 𝑎 1 𝑛 𝑚 = 𝑛 𝑎 𝑚 = 𝑎 𝑚 𝑛

20. Rational Exponents With Denominator Other Than 1
Since multiplication is commutative, we have 𝑎 𝑚 1 𝑛 = 𝑎 𝑚⋅ 1 𝑛 = 𝑎 𝑚 𝑛
But this can also be written 𝑎 𝑚 1 𝑛 = 𝑛 𝑎 𝑚 = 𝑎 𝑚 𝑛
In general, 𝑎 𝑚 𝑛 = 𝑛 𝑎 𝑚 = 𝑛 𝑎 𝑚
Note that the denominator becomes the index and the numerator is an exponent in the radical or outside the radical (both are correct)

21. Rational Exponents With Denominator Other Than 1
Example: rewrite the expression with rational exponents as a radical expression
= = = 3 16 ≈2.520
= = = 4 32 ≈2.378
− = 5 −3 3 = 5 −3 3 = 5 −27 ≈−1.933

22. Guided Practice
Rewrite the expression as a radical, then evaluate using a calculator. = =
− =
− =

23. Guided Practice
Rewrite the expression as a radical, then evaluate using a calculator.
= = 3 2 =9
= = 2 3 =8
− = 3 − = −4 4 =256
− = 5 −9 3 ≈−3.737

24. Negative Rational Exponents
We defined negative exponents to mean 𝑥 −𝑛 = 𝑥 −𝑛 1 = 1 𝑥 𝑛
That is, you can move the numerator to the denominator and change the sign of the exponent
We can understand negative rational exponents using this definition 𝑥 − 𝑚 𝑛 = 1 𝑥 𝑚 𝑛 = 1 𝑛 𝑥 𝑚 = 1 𝑛 𝑥 𝑚 , 𝑥≠0

25. Negative Rational Exponents
Some examples
8 − = = = = 1 16
−64 − = 1 − = − = 1 −4 2 = 1 16
4 − = = = ≈0.574

26. Guided Practice
Rewrite the expression as a radical, then evaluate using a calculator.
−30 − =
9 − =
243 − =

27. Guided Practice
Rewrite the expression as a radical, then evaluate using a calculator.
−30 − = − = ≈0.104
9 − = = = ≈
243 − = = = 1 9

28. Solving Radical Equations
We were able to use the square root property to solve equations
We can also use the nth root definition to solve equations
In general, if 𝑥 𝑛 =𝑎, then 𝑥= 𝑛 𝑎
Some examples
4 𝑥 5 =128⟹ 𝑥 5 =32⟹𝑥= 5 32 =2
𝑥−3 4 =21⟹𝑥−3=± 4 21 ⟹3± 𝑥= ≈5.141, 𝑥=3− 4 21 ≈0.859

29. Guided Practice
Solve the equation. Round the result to three decimal places when appropriate.
𝑥 3 =64
1 2 𝑥 5 =512
𝑥−2 3 =−14

30. Guided Practice
Solve the equation. Round the result to three decimal places when appropriate.
𝑥 3 =64⟹𝑥= 3 64 =4
1 2 𝑥 5 =512⟹ 𝑥 5 =1024⟹𝑥= =4
𝑥−2 3 =−14⟹𝑥−2= 3 −14 ⟹𝑥=2+ 3 −14 ≈−0.410

31. Exercise 6.1
Handout